In this letter, a computational approach for solving cross-coupled algebraic Riccati equations (CAREs) is investigated. The main purpose of this letter is to propose a new algorithm that combines Newton's method with a gradient-based iterative (GI) algorithm for solving CAREs. In particular, it is noteworthy that both a quadratic convergence under an appropriate initial condition and reduction in dimensions for matrix computation are both achieved. A numerical example is provided to demonstrate the efficiency of this proposed algorithm.